286 X EXISTENCE THEOREMS

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and similarly

t7(0 - t?(0) - | f(s, i?(s)) ds
whence

\\u(t) — v(t)\\ ^ ||w(0) — 0(0)|| + (/(51, u(sj) —f(s, v(s))) ds +(«! + ;
From the assumption on D2/and from (8.5.4) and (8.7.7) this yields
(10.5.1.2.) w(0 ^ w(0) + (8j + e2)r + k

=r-

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where w(t) = \\u(t) - v(t)\\. Theorem (10.5.1) is then a consequence of the
following lemma:

(10.5.1.3) (Gronwall's lemma) If, in an interval [0, c], (p and \j/ are two
regulated functions ^ 0, then for any regulated function w ^ 0 in [0, c] satisfying
the inequality

(10.5.1.4)

we /zat>e /« [0, c]

(10.5.1.5)

• I m

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w(t) < <p(t) + \l/(s)w(s) ds

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( <

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s) exp

ds.

Write .y(O) = il/(s)w(s) ds; y is continuous, and from (10.5.1.4) it follows

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that, in the complement of a denumerable subset of [0, c], we have

(10.5.1.6) /(0-WOXO<?W(0

by Section 8.7. Write z(0 = XOexp(- fVfa)*); relation (10.5.1.6) is

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equivalent to

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By (8.5.3) and using the fact that z(G) = 0, we get, for t e [0, c]

f / r \

z(0 ^ I <9(s)^(s) expl — I ^(£) rf^) ds
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whence by definition

!f

•/(

and (10.5.1.5) now follows from the relation w(t) < ^>(r) + y(t).pose || D2 /(/,*) || < k In I x H. If u, v are two approximate